Stuart Riffle (2011) argues that visualization is the first step towards understanding concepts such as the Discrete Fourier Transform (DFT): "Once you can 'see' what’s going on in your head, a lot of the quirks of working with the DFT become much less mysterious." His one sentence description for the Discrete Fourier Transform below is succinct and instructive, albeit purely symbolic.
Follow the link above to read the rest of Stuart Riffle's article, and especially his visualizations. As with any language, mathematics is first and foremost about visualization followed by the algebraic (symbolic) expressions. Those who begin with the algebraic (symbolic) expressions tend to get lost along the way.
Source: Riffle, S (2011, May 17), Understanding the Fourier Transform, #AltDevBlog.